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Error Detection Schemes for τ NAF Conversion within Koblitz Curves Benchmarked on Various ARM Processors
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  • Kasra Ahmadi ,
  • Saeed Aghapour ,
  • Mehran Mozaffari Kermani ,
  • Reza Azarderakhsh
Kasra Ahmadi
University of South Florida

Corresponding Author:[email protected]

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Saeed Aghapour
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Mehran Mozaffari Kermani
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Reza Azarderakhsh
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Abstract

Elliptic curves cryptography, the most widely-deployed pre-quantum public key cryptography, can be implemented efficiently with Koblitz curves. The reason for these realizations is that in such efficient architectures, through using Frobenius endomorphism, the high cost of doubling can be ameliorated by simple shifting. However, in order to use this property, scalars should be represented by a τ-expansion. Such curves require integer to τ-NAF conversion, which is a prominent factor in Koblitz curves cryptography. Nevertheless, natural and malicious faults, can threaten the reliability of such constructions. In cryptosystems, verifying the correctness of the underlying computation implemented in hardware and software platforms is extremely important to detect permanent and transient errors. In this paper, for the first time to the best of our knowledge, we investigate fault detection schemes in single and double τ-NAF (nonadjacent form) conversion algorithms. To this end, we propose refined algorithms and implementation to resist both permanent and transient error occurrence using a number of fault models to make sure the performed assessments reflect the results accurately. Additionally, we simulate the proposed algorithms in Python environment with single, random, and burst fault models resulting in very high error coverage. Finally, we implement our scheme on ARMv7 and ARMv8 platforms to show the overhead of our implementation. We achieved less than 17% clock cycle overhead on Cortex-M4 and about 25% on Cortex-A72 processors. Our proposed scheme code size overhead was less than 6%. The proposed approaches make the implementations of Koblitz curves τ-NAF conversion more reliable with acceptable overheads.